# AL-KHWARIZMI (800-847)

AL-KHWARIZMI

The man often credited with the introduction of ‘Arabic’ numerals was al-Khwarizmi, an Arabian mathematician, geographer and astronomer. Strictly speaking it was neither invented by al-Khwarizmi, nor was it Middle Eastern in origin.

786 – Harun al-Rashid came to power. Around this time al-Khwarizmi born in Khwarizm, now Khiva, in Uzbekistan.

813 – Caliph al-Ma’mun, the patron of al-Khwarizmi, begins his reign in Baghdad.

Arabic notation has its roots in India around 500 AD, thus the current naming as the ‘Hindu-Arabic’ system. al-Khwarizmi, a scholar in the Dar al-ulum (House of Wisdom) in Baghdad in the ninth century, is responsible for introducing the numerals to Europe. The method of using only the digits 0-9, with the value assigned to them determined by their position, as well as introducing a symbol for zero, revolutionised mathematics.

al-Khwarizmi explained how this system worked in his text ‘Calculation with Hindu numerals‘. He was clearly building upon the work of others before him, such as DIOPHANTUS and BRAHMAGUPTA, and on Babylonian sources that he accessed through Hebrew translations. By standardizing units, Arabic numerals made multiplication, division and every other form of mathematical calculation much simpler. His text ‘al-Kitab al-mukhtasar- fi hisab al-jabr w’al-muqabala’ (The Compendious Book on Calculating by Completion and Balancing) gives us the word algebra. In this treatise, al-Khwarizmi provides a practical guide to arithmetic.

In his introduction to the book he says the aim of the work is to introduce ‘what is easiest and most useful in mathematics, such as men constantly require in cases of inheritance, legacies, partition, lawsuits and trade, and in all their dealings with one another, or when measuring lands, digging canals and making geometrical calculations.’ He introduced quadratic equations, although he described them fully in words and did not use symbolic algebra.
It was in his way of handling equations that he created algebra.

The two key concepts were the ideas of completion and balancing of equations. Completion (al-jabr) is the method of expelling negatives from an equation by moving them to the opposite side

4x2 = 54x – 2x2  becomes  6x2 = 54x

Balancing (al-muqabala) meanwhile, is the reduction of common positive terms on both sides of the equation to their simplest forms

x2 + 3x + 22 = 7x + 12  becomes  x2 + 10 = 4x

Thus he was able to reduce every equation to simple, standard forms and then show a method of solving each, showing geometrical proofs for each of his methods – hence preparing the stage for the introduction of analytical geometry and calculus in the seventeenth century.

The name al-Khwarizmi also gives us the word algorithm meaning ‘a rule of calculation’, from the Latin title of the book, Algoritmi de numero Indorum.

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# IBN SINA (AVICENNA) (980-1037)

‘al Qann fi al-Tibb’ (The Canon of Medicine), also ‘ The Book of the Remedy

Avicenna lived under the Sammarid caliphs in Bukhara. He identified different forms of energy – heat, light and mechanical – and the idea of a force.

AVICENNA

Before GALEN, scientists describing nature followed the old Greek traditions of giving the definitions and following them up with the body of logical development. The investigator was then obliged merely to define the various types of ‘nature’ to be found. With Galen this procedure was changed.

Instead of hunting for these natures and defining more and more of them, reproducing ARISTOTLE’s ideas, AVICENNA, a Persian physician, planned inductive and deductive experimental approaches to determine the conditions producing observable results.

His tome surveyed the entire field of medical knowledge from ancient times up to the most up to date Muslim techniques. Avicenna was the first to note that tuberculosis is contagious; that diseases can spread through soil and water and that a person’s emotions can affect their state of physical health. He was the first to describe meningitis and realize that nerves transmit pain. The book also contained a description of 760 drugs. Its comprehensive and systematic approach meant that once it was translated into Latin in the twelfth century it became the standard medical textbook in Europe for the next 600 years.

Arabic Canon of Medicine by Avicenna 1632

Avicenna thought of light as being made up of a stream of particles, produced in the Sun and in flames on Earth, which travel in straight lines and bounce off objects that they strike.

A pinhole in a curtain in a darkened room causes an inverted image to be projected, upside-down, onto a wall opposite the curtained window. The key point is that light travels in straight lines. A straight line from the top of a tree some distance away, in a garden that the window of the camera obscura faces onto – passing through the hole in the curtain – will carry on down to a point near the ground on the wall opposite. A straight line from the base of the tree will go upwards through the hole to strike the wall opposite near the ceiling. Straight lines from every other point on the tree will go through the hole to strike the wall in correspondingly determined spots, the result is an upside-down image of the tree (and of everything else in the garden).

He realized that refraction is a result of light traveling at different speeds in water and in air.

He used several logical arguments to support his contention that sight is not a result of some inner light reaching outward from the eye to probe the world around it, but is solely a result of light entering the eye from the world outside – realizing that ‘after-images’ caused by a bright light will persist when the eyes are closed and reasoning that this can only be the result of something from outside affecting the eyes. By effectively reversing the extro-missive theory of Euclid, he formulated the idea of a cone emanating from outside the eye entering and thus forming an image inside the eye – he thus introduced the modern idea of the ray of light.

The idea which was to have the most profound effect on the scientific development of an understanding of the behaviour of light was the thought of the way images are formed on a sunny day by the ‘camera obscura’.

## AL HAZEN (c.965-1039)

Born in Basra and working in Egypt under al-Hakim, Abu Ali al-Hassan ibn al-Haytham was one of the three greatest scientists of Islam (along with al-Biruni and ibn-Sina). He explained how vision works in terms of geometric optics and had a huge influence on Western science. He is regarded as one of the earliest advocates of the scientific method.

The mathematical technique of ‘casting out of nines’, used to verify squares and cubes, is attributed to al-Hazen.

Al-Hazen dissented with the J’bir Ayam hypothesis of transmutation, thus providing two different strands for Alchemy in Europe from the Islāmic world.

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MEDICINE

# LEONARDO FIBONACCI (c.1170-c.1250)

#### Also known as Leonardo Pisano. Published ‘Liber Abaci’ in 1202.

1202 – Italy

FIBONACCI

FIBONACCI

‘A series of numbers in which each successive term is the sum of the preceding two’

For example:   1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , 144….

The series is known as the Fibonacci sequence and the numbers themselves as the Fibonacci numbers.

The Fibonacci sequence has other interesting mathematical properties – the ratio of successive terms ( larger to smaller;   1/1, 2/1, 3/2, 5/3, 8/5…. ) approaches the number 1.618
This is known as the golden ratio and is denoted by the Greek letter Phi.

Phi was known to ancient Greeks.
Greek architects used the ratio 1:Phi as part of their design, the most famous example of which is the Parthenon in Athens.

Fibonacci sequence in flower petals

Phi also occurs in the natural world.
Flowers often have a Fibonacci number of petals.

During his travels in North Africa, Fibonacci learned of the decimal system of numbers that had evolved in India and had been taken up by the Arabs.
In his book Liber Abaci he re-introduced to Europe the Arabic numerals that we use today, adhering roughly to the recipe ‘the value represented must be proportional to the number of straight lines in the symbol’.

Following the Arabs, Fibonacci ( ‘son of the simpleton’ euph. or ‘son of the innocent’ ) introduced the place–value concept, with each position representing a different power of ten and these arranged in ascending order from right to left.

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# RENE DESCARTES (1596-1650)

1637 – France

Cogito ergo sum‘ – The result of a thought experiment resolving to cast doubt on any and all of his beliefs, in order to discover which he was logically justified in holding.

Descartes argued that although all his experience could be the product of deception by an evil daemon, the demon could not deceive him if he did not exist.

His theory that all knowledge could be gathered in a single, complete science and his pursuit of a system of thought by which this could be achieved left him to speculate on the source and the truth of all existing knowledge. He rejected much of what was commonly accepted and only recognised facts that could intuitively be taken as being beyond any doubt.

His work ‘Meditations on First Philosophy’ (1641) is centered on his famous maxim. From this he would pursue all ‘certainties’ via a method of systematic, detailed mental analysis. This ultimately led to a detached, mechanistic interpretation of the natural world, reinforced in his metaphysical text ‘Principia Philosophiae‘ (1644) in which he attempted to explain the universe according to the single system of logical, mechanical laws he had earlier envisaged and which, although largely inaccurate, would have an important influence even after Newton. He envisaged the human body as subject to the same mechanical laws as all matter; distinguished only by the mind, which operated as a distinct, separate entity.

Through his belief in the logical certainty of mathematics and his reasoning that the subject could be applied to give a superior interpretation of the universe came his 1637 appendix to the ‘Discourse’, entitled ‘La Geometrie‘, Descartes sought to describe the application of mathematics to the plotting of a single point in space.

This led to the invention of ‘Cartesian Coordinates’ and allowed geometric expressions such as curves to be written for the first time as algebraic equations. He brought the symbolism of analytical geometry to his equations, thus going beyond what could be drawn. This bringing together of geometry and algebra was a significant breakthrough and could in theory predict the future course of any object in space given enough initial knowledge of its physical properties and movement.

Descartes showed that circular motion is in fact accelerated motion, and requires a cause, as opposed to uniform rectilinear motion in a straight line that has the property of inertia – and if there is to be any change in this motion a cause must be invoked.

By the 1660s, there were two rival theories about light. One, espoused by the French physicist Pierre Gassendi (1592-1655) held that it was a stream of tiny particles, traveling at unimaginably high-speed. The other, put forward by Descartes, suggested that instead of anything physically moving from one place to another the universe was filled with some material (dubbed ‘plenum’), which pressed against the eyes. This pressure, or ‘tendency of motion’, was supposed to produce the phenomenon of sight. Some action of a bright object, like the Sun, was supposed to push outwards. This push was transmitted instantaneously, and would be felt by the human eye looking at a bright object.

There were problems with these ideas. If light is a stream of tiny particles, what happens when two people stand face-to-face looking each other in the eye? And if sight is caused by the pressure of the plenum on the eye, then a person running at night should be able to see, because the runner’s motion would make the plenum press against their eyes.

Descartes original theory is only a small step to a theory involving pulses of pressure spreading out from a bright object, like the pulses of pressure that would travel through water if you slap the surface, and exactly equivalent to pressure waves which explain how sound travels outward from its source.

TIMELINE

# PIERRE DE FERMAT (1601- 65) ANDREW WILES (b.1953)

1637 – France; 1993 – USA

PIERRE DE FERMAT

Fermat’s theorem proves that there are no whole-number solutions of the equation x n + y n = z n for n greater than 2

The problem is based on Pythagoras’ Theorem; in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares on the other two sides; that is x 2 + y 2 = z 2

If x and y are whole numbers then z can also be a whole number: for example 52+ 122 = 132
If the same equation is taken to a higher power than 2, such as x 3 + y 3 = z 3 then z cannot ever be a whole number.

In about 1637, Fermat wrote an equation in the margin of a book and added ‘I have discovered a truly marvelous proof, which this margin is too small to contain’. The problem now called Fermat’s Last Theorem baffled mathematicians for 356 years.

ANDREW WILES

In 1993, Wiles, a professor of mathematics at Princeton University, finally proved the theorem.

Wiles, born in England, dreamed of proving the theorem ever since he read it at the age of ten in his local library. It took him years of dedicated work to prove it and the 130-page proof was published in the journal ‘Annals of Mathematics‘ in May 1995.

TIMELINE

MATHEMATICS

# BLAISE PASCAL (1623- 62)

1647 – France

BLAISE PASCAL

‘When pressure is applied anywhere to an enclosed fluid, it is transmitted uniformly in all directions’

EVANGELISTA TORICELLI (1608-47) had argued that air pressure falls at higher altitudes.

Using a mercury barometer, Pascal proved this on the summit of the 1200m high Puy de Dome in 1647. His studies in this area led to the development of PASCAL’S PRINCIPLE, the law that has practical applications in devices such as the car jack and hydraulic brakes. This is because the small force created by moving a lever such as the jacking handle in a sizable sweep equates to a large amount of pressure sufficient to move the jack head a few centimetres.
The unit of pressure is now termed the pascal.

‘The study of the likelihood of an event’

Together with PIERRE DE FERMAT, Pascal developed the theory of probabilities (1654) using the now famous PASCAL’S TRIANGLE.

Chance is something that happens in an unpredictable way. Probability is the mathematical concept that deals with the chances of an event happening.

Probability theory can help you understand everything from your chances of winning a lottery to your chances of being struck by lightning. You can find the probability of an event by simply dividing the number of ways the event can happen by the total number of possible outcomes.
The probability of drawing an ace from a full pack of cards is 4/52 or 0.077.

Probability ranges from 1 (100%) – Absolutely certain, through Very Likely 0.9 (90%) and Quite Likely 0.7 (70%), Evens (Equally Likely) 0.5 (50%), Not Likely 0.3 (30%) and Not Very Likely 0.2 (20%), to Never – Probability 0 (0%).

The computer language Pascal is named in recognition of his invention in 1644 of a mechanical calculating machine that could add and subtract.

Like many of his contemporaries, Pascal did not separate philosophy from science; in his book ‘Pensees’ he applies his mathematical probability theory to the problem of the existence of God. In the absence of evidence for or against God’s existence, says Pascal, the wise man will choose to believe, since if he is correct he will gain his reward, and if he is incorrect he stands to lose nothing.

TIMELINE

COMPUTERS

# GOTTFRIED LEIBNIZ (1646-1716)

1684 – Germany

‘A new method for maxima and minima, as well as tangents … and a curious type of calculation’

Newton invented calculus (fluxions) as early as 1665, but did not publish his major work until 1687. The controversy continued for years, but it is now thought that each developed calculus independently.
Terminology and notation of calculus as we know it today is due to Leibniz. He also introduced many other mathematical symbols: the decimal point, the equals sign, the colon (:) for division and ratio, and the dot for multiplication.

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MECHANICS